Exercises to Hand In

1.

Draw a gradient/contour plot for the function $z=x\exp(-x^{2}-y^{2})$ in the rectangle $-2\leq x\leq2,\,$ $-2\leq y\leq2$ (usually written as simply $-2\leq x,\,y\leq2$). Use this to locate, approximately, the critical points. Classify them: max, min or saddle; explain how you know.

2.

Do the same as exercise 1, except use the rectangle $-10\leq x,y\leq10$and the function:

z=4500-105x²-105y²+3y²x+3x²y+0.8x⁴+0.8y⁴.

3.

Do the same as exercise 1, except use the rectangle $0\leq x,y\leq\pi$and the function: $z=.02\sin(x)\sin(y)-.03\sin(2x)\sin(y)+.04\sin (x)\sin(2y)+.08\sin(2x)\sin(2y)$ .

(see exercise 3 of the previous set for a sample m-file).

4.

Do the same as exercise 1, except use the rectangle $-\pi\leq x,y\leq \pi$ and the function:

$z=\cos(x+y)\cos(3x-y)+\cos(x-y)\sin(x+3y)+5\exp(-(x^{2}+y^{2})/8)$

(see exercise 3 of the previous set for a sample m-file).